See the double copper clip at the base of the motor where you attach the two horn wires on either side of it? That has to be electrically isolated from the body of the horn. If that clip is making contact with the horn body, it will run the horn motor when you attach it to the light bar. As you say, completing the circuit.
Get your voltmeter out and do a continuity test between the double copper clip and horn body. There should be NO continuity. There should be NO continuity between the tubular rivet and copper clip that holds it in place. The rubber washer under the rivet insulates the clip from the rivet. Try that test and get back to us on your results. Good luck.